Question

Let n be in N and let K be a field. Show that for a linear map T
: K^{n} to K^{n} the following statements are
equivalent:

1. The map T is one-to-one (injective).

2. The map T is onto (surjective).

3. The map T is invertible.

4. The map T is an isomorphism.

Answer #1

Problem 3. Recall that a linear map f : V → W is called an
isomorphism if it is invertible (i.e. has a linear inverse map). We
proved in class that f is in fact invertible if and only if it is
bijective. Use this fact from class together with the Rank-Nullity
Theorem (of the previous problem) to show that if f : V → V is an
endomorphism, then it is actually invertible if
1. it is merely injective...

a. Let T : V → W be left invertible. Show that T is
injective.
b. Let T : V → W be right invertible. Show that T is
surjective

True or False? No reasons needed.
(e) Suppose β and γ are bases of F n and F m, respectively.
Every m × n matrix A is equal to [T] γ β for some linear
transformation T: F n → F m.
(f) Recall that P(R) is the vector space of all polynomials with
coefficients in R. If a linear transformation T: P(R) → P(R) is
one-to-one, then T is also onto.
(g) The vector spaces R 5 and P4(R)...

Let A be an n by n matrix, with real
valued entries. Suppose that A is NOT invertible.
Which of the following statements are true?
?Select ALL correct answers.?
The columns of A are linearly dependent.
The linear transformation given by A is one-to-one.
The columns of A span
Rn.
The linear transformation given by A is onto
Rn.
There is no n by n matrix D such that
AD=In.
None of the above.

Let p(x) be an irreducible polynomial of degree n over a finite
field K. Show that its Galois group over K is cyclic of order n and
then show how the Galois group of x3 − 1 over Q is
cyclic of order 2.

(Linear Algebra)
A n×n-matrix is nilpotent if there is a "r" such that
Ar is the nulmatrix.
1. show an example of a non-trivial, nilpotent 2×2-matrix
2.let A be an invertible n×n-matrix. show that A is not
nilpotent.

n x n matrix A, where n >= 3. Select 3 statements from the
invertible matrix theorem below and show that all 3 statements are
true or false. Make sure to clearly explain and justify your
work.
A=
-1 , 7, 9
7 , 7, 10
-3, -6, -4
The equation A has only the trivial solution.
5. The columns of A form a linearly independent set.
6. The linear transformation x → Ax is one-to-one.
7. The equation Ax...

Given that A and B are n × n matrices and T is a linear
transformation. Determine which of the following is FALSE.
(a) If AB is not invertible, then either A or B is not
invertible.
(b) If Au = Av and u and v are 2 distinct vectors, then A is not
invertible.
(c) If A or B is not invertible, then AB is not invertible.
(d) If T is invertible and T(u) = T(v), then u =...

1. Let V and W be ﬁnite-dimensional vector spaces over ﬁeld F
with dim(V) = n and dim(W) = m, and
let φ : V → W be a linear transformation.
A) If m = n and ker(φ) = (0), what is im(φ)?
B) If ker(φ) = V, what is im(φ)?
C) If φ is surjective, what is im(φ)?
D) If φ is surjective, what is dim(ker(φ))?
E) If m = n and φ is surjective, what is ker(φ)?
F)...

Let V and W be vector spaces and let T:V→W be a linear
transformation. We say a linear transformation S:W→V is a left
inverse of T if ST=Iv, where ?v denotes the identity transformation
on V. We say a linear transformation S:W→V is a right inverse of ?
if ??=?w, where ?w denotes the identity transformation on W.
Finally, we say a linear transformation S:W→V is an inverse of ? if
it is both a left and right inverse of...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 3 minutes ago

asked 13 minutes ago

asked 17 minutes ago

asked 18 minutes ago

asked 19 minutes ago

asked 22 minutes ago

asked 22 minutes ago

asked 22 minutes ago

asked 22 minutes ago

asked 28 minutes ago

asked 50 minutes ago

asked 59 minutes ago